Which expression represents the formal derivative at a point x using the limit definition?

Which expression represents the formal derivative at a point x using the limit definition?

• A. f'(x) = lim_{h->0} (f(x+h) − f(x))/h
• B. f'(x) = lim_{h->0} (f(x) − f(x-h))/h
• C. f'(x) = lim_{h->0} (f(x+h) − f(x-h))/ (2h)
• D. f'(x) = lim_{h->0} (f(x) − f(x))/h

Answer: (A)

The derivative at a point is defined as the limit of the slope of the secant line as the increment h goes to zero: f'(x) = lim_{h->0} [f(x+h) - f(x)]/h. This forward difference form directly measures how f changes over a tiny step to the right of x, and as h gets vanishingly small it becomes the instantaneous rate of change. This expression is the standard, canonical limit definition because it literally computes the average rate of change over the interval [x, x+h] and lets the interval shrink to a point. The other forms can also yield the derivative under the usual differentiability assumption, but they aren’t the textbook definition. The backward-difference form, with f(x) - f(x-h) over h, is equivalent to the forward form once you substitute h with -h and take the limit. The central-difference form, (f(x+h) - f(x-h)) / (2h), also tends to f'(x) as h -> 0 if the derivative exists, since it averages slopes from both sides. The last option, (f(x) - f(x))/h, is always 0 for h ≠ 0, so its limit would be 0, not the general derivative. So the forward difference expression is the direct, standard way to represent the formal derivative.

The derivative at a point is defined as the limit of the slope of the secant line as the increment h goes to zero: f'(x) = lim_{h->0} [f(x+h) - f(x)]/h. This forward difference form directly measures how f changes over a tiny step to the right of x, and as h gets vanishingly small it becomes the instantaneous rate of change.

This expression is the standard, canonical limit definition because it literally computes the average rate of change over the interval [x, x+h] and lets the interval shrink to a point. The other forms can also yield the derivative under the usual differentiability assumption, but they aren’t the textbook definition. The backward-difference form, with f(x) - f(x-h) over h, is equivalent to the forward form once you substitute h with -h and take the limit. The central-difference form, (f(x+h) - f(x-h)) / (2h), also tends to f'(x) as h -> 0 if the derivative exists, since it averages slopes from both sides. The last option, (f(x) - f(x))/h, is always 0 for h ≠ 0, so its limit would be 0, not the general derivative.

So the forward difference expression is the direct, standard way to represent the formal derivative.